Problem 22 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(y=-3 x+4\)

Step-by-Step Solution

Answer

The equation in standard form is \(3x + y = 4\) with \( A = 3 \), \( B = 1 \), and \( C = 4 \).

Step 1: Understand the Standard Form

The standard form of a linear equation in two variables is given by \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be non-negative.

Step 2: Rearrange Equation to Match Standard Form

Start with the given equation \( y = -3x + 4 \). We need to rearrange it to the form \( Ax + By = C \). The first step is to move \(-3x\) to the left side of the equation by adding \(3x\) to both sides. This gives us \( 3x + y = 4 \).

Step 3: Identify A, B, and C

Compare the rearranged equation \( 3x + y = 4 \) to the standard form \( Ax + By = C \). You'll see that \( A = 3 \), \( B = 1 \), and \( C = 4 \).

Key Concepts

Linear Equations

Linear equations form the basis for many algebraic calculations and are essential for understanding the relationships between variables. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The standard form of a linear equation in two variables is written as \( Ax + By = C \). Here:

• \(A\), \(B\), and \(C\) are constants or numbers.
• \(x\) and \(y\) are variables.

Linear equations typically represent a straight line when graphed, and they are called "linear" because of the constant rate of change between the variables. This concept is crucial in understanding more complex algebraic and calculus concepts. Remember, linear equations help us model real-world relationships and solve problems ranging from basic calculations to complex data analysis.

Integer Coefficients

In the context of linear equations, understanding integer coefficients is crucial. Coefficients are the numbers in front of the variables in an equation. For the standard form \(Ax + By = C\):

• \(A\) and \(B\) are the coefficients placed before the variables \(x\) and \(y\).
• These coefficients are ideally integers, which means they are whole numbers, not fractions or decimals.

Using integer coefficients has practical benefits:

• Simplicity: Integer coefficients make equations easier to manipulate and solve.
• Universality: They provide a standard format that is widely understood and used by mathematicians.

Having integer coefficients aligns with the requirement for the equation to be in a "standard" form, ensuring it is consistent and clear for anyone analyzing or solving it.

Rearranging Equations

Rearranging equations is a skills-based task crucial for transforming equations into standard form. This process involves manipulating the equation while maintaining its equality to arrive at a different, often simpler or more useful, format.Let's go through it with an example, using the equation \(y = -3x + 4\):

• Our goal is to move all terms involving variables \(x\) and \(y\) to one side, aiming for the form \(Ax + By = C\).
• Start by adding \(3x\) to both sides of the equation to shift \(x\) terms with \(y\). This gives \(3x + y = 4\).

This rearrangement helps reveal:

• The equation is now in standard form.
• The values of \(A\), \(B\), and \(C\) can be easily identified relative to standard form.

Rearranging makes the equation more intuitive to interpret and facilitates comparisons, calculations, and graphical interpretations.